Yep, you're close to the answer, except two things. First, substract the lower temperature from the higher temperature. Second, your mistake, after the ice has melted don't use c(ice) any more, but c(water).

Ok, it depends on the context where you use it. I was thinking that you were using the equality [tex]Q_{gained} = Q_{lost}[/tex], meaning that the heat gained by ice is equal to the heat lost by water. T_final - T_initial would make the RHS negative.
On the other hand, if you use [tex]Q_{gained} + Q_{lost} = 0[/tex] it is ok to use as you did, assuming that the lost energy is negative.